Chapter 3 Geometry of convex functions - Meboo Publishing ...

226 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.3.2.1.3 Exercise. k-largest norm gradient.
Prove (510). Find ∇‖x‖ 1 and ∇‖x‖n
k
on R n . 3.11
3.3.3 clipping
Zeroing negative vector entries under 1-norm is accomplished:
‖x + ‖ 1 = minimize 1 T t
t∈R n
subject to x ≼ t
0 ≼ t
(512)
where, for x=[x i , i=1... n]∈ R n
x + t ⋆ =
(clipping)
[
xi , x i ≥ 0
0, x i < 0
}
, i=1... n
]
= 1 (x + |x|) (513)
2
minimize
x∈R n ‖x + ‖ 1
subject to x ∈ C
≡
minimize 1 T t
x∈R n , t∈R n
subject to x ≼ t
0 ≼ t
x ∈ C
(514)
3.4 Inverted functions and roots
A given function f is convex iff −f is concave. Both functions are loosely
referred to as convex since −f is simply f inverted about the abscissa axis,
and minimization of f is equivalent to maximization of −f .
A given positive function f is convex iff 1/f is concave; f inverted about
ordinate 1 is concave. Minimization of f is maximization of 1/f .
We wish to implement objectives of the form x −1 . Suppose we have a
2×2 matrix
[ ] x z
T ∈ R 2 (515)
z y
3.11 Hint:D.2.1.